I compare the hierarchical risk parity (HRP) portfolio to six other portfolio strategies: inverse volatility (IV), equal weight (EW), global minimum variance (GMV), maximum diversification (MD) equal risk contribution (ERC) and maximum sharpe ratio (MSR). I report out-of sample performance, portfolio composition and correlation in two asset sets: S&P500 stocks from Dec 1991 thru Nov 2016 and in global futures from Jan 2002 thru Apr 2016. I simulate both the original version of the HRP portfolio using variance risk measure (HRP-VAR) as well as a similarly defined portfolio using the volatility risk measure (HRP-VOL).
HRP-VOL turns out to be very closely related the ERC portfolio. Performances are highly correlated, and both portfolios allocate funds in similar proportions to assets and reach a similar performance. Both are essentially mechanisms to allocate a share of volatility risk to every asset, leading to the same benefits (balanced allocation, smaller dependence on estimation error) and drawbacks (sub-optimal diversification). Furthermore, the benefits of the HRP portfolio claimed by the author - reduced sensitivity to parameter estimation leading to increased robustness in large asset sets - apply to both strategies and do not discriminate between them.
HRP-VAR, the original hierarchical risk parity portfolio based on variance risk, outperforms its declared benchmark, the GMV portfolio, in global futures. This is of limited importance, as the GMV portfolio performs poorly in this data set in the first place and since it concentrates the allocation in a small number of assets with lowest variance. The outperformance is caused by a a more balanced allocation of the HRP-VAR portfolio. I imagine that an ERC portfolio based on the variance risk measure would lead to a similar outcome.
While I sympathize with the algorithmic approach of the portfolio, I do not see a clear benefit over the already established and much better researched alternatives.
López de Prado (2016) orders assets hierarchically and then recursively cuts asset in halves, allocating an equal share of risk to both halfs.
He claims that this algorithm reduces sensitivity to paramter estimation, leads to more stable portfolios and is particularely well suited for a large number of portfolio constituents.
López de Prado (2016) uses the variance risk measure \(\sigma^2\) and uses the global minimum variance (GMV) portfolio as a benchmark. I name his definition HRP-VAR and introduce a similar portfolio, HRP-VOL, based on volatility instead of variance. Risk is now measured as \(\sigma = \sqrt{\mathbf{w'\Sigma w}}\) using \(w_i = 1 / \sigma_i\).
Figures 1-4 show the dendrogram of hierarchical clusters computed with global futures at four moments throughout our study period. Clusters are based on the previous 3 years of returns. Colors assign futures instruments to asset classes equities, commodities and fixed income.
Observations:
Figure 1: Hierarchical clustering of Global Futures as of 2002-01-31 based on three years of daily returns.
Figure 2: Hierarchical clustering of Global Futures as of 2006-10-31 based on three years of daily returns.
Figure 3: Hierarchical clustering of Global Futures as of 2011-07-29 based on three years of daily returns.
Figure 4: Hierarchical clustering of Global Futures as of 2016-04-29 based on three years of daily returns.
\[ \min_{w \geq 0} \mathbf{w}'\mathbf{P}\mathbf{w} \]
The inverse volatility (IV) portfolio scales assets to equal single-asset volatility. \[ w_i = \frac{1}{\sigma_r} \]
The maximum diversified (MD) portfolio proposed by Choueifaty and Coignard (2008) maximizes the diversification ratio defined as the ratio between weighted sum of single asset volatility over portfolio volatility. \[ \max_{\mathbf{w}\geq 0} \text{DR}, \quad \text{DR} = \frac{\textbf{w}'R(\textbf{r})}{R(\textbf{w}'\textbf{r})} = \frac{\mathbf{w}'\mathbf{\sigma}_r}{\sqrt{\mathbf{w}'\mathbf{\Sigma}\mathbf{w}}} \]
The equal risk contribution (ERC) portfolio formalized by Maillard, Roncalli, and Teïletche (2010) aligns the contribution of each asset to the total portfolio risk. \[ \text{RC}_j = \text{RC}_i = w_i\frac{\partial R(\mathbf{w}'\mathbf{r})}{\partial w_i} = \frac{1}{2} w_i (\mathbf{\Sigma}\mathbf{w})_i \]
The equal weight (EW) portfolio allocates an equal amount to all assets: \[ w_i = \frac{1}{N} \]
The Maximum sharpe ratio (MSR) portfolio maximizes the portfolio’s excess return divided by volatility: \[ \max_{w\geq 0} \frac{\mathbf{w}'\;\mathbb{E}\left[\mathbf{r} - r_f\right]}{\sqrt{\mathbf{w}'\Sigma\mathbf{w}}} \]
Portfolios are calibrated subject to two alternative investment constraints:
Figures 5-8 and tables 3-5 report out-of-sample risk and performance of the portfolio strategies in both asset sets. Portfolios are calibrated monthly based on the last three years of underlying market returns. Stock returns are reported in excess of the 6-monht T-bill rate. Futures returns are considered to be excess returns. Execution costs are not taken into account.
Observations:
Figure 5: Rolling one-year out-of-sample return and volatility. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.
Table 1: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.
Figure 6: Rolling one-year out-of-sample volatility and return in excess of the 6 month T-bill rate. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.
Table 2: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is in excess of the 6-month T-bill rate. Execution cost is not taken into account. MSE reports the mean squared difference to the 10% volatility target in above figure.
Figure 7: Rolling one-year out-of-sample volatility and return in excess of the 6 month T-bill rate. Execution cost is not taken into account. The plot is interactive, lines can be removed/restored by selecting the legend.
Table 3: Out-of-sample risk/performance statistics based on weekly returns from Jan 2002 thru Apr 2016. Annual return is reported in excess of the 6-month T-bill rate. Execution cost is not taken into account.
Figures 8-10 below report the pairwise correlation coefficients between out-of-sample returns of the simulated portfolio strategies over the entire study period.
Observations:
Figure 8: Pairwise correlation between daily returns of portfolio strategies from Jan 2002 thru Apr 2016 in Global Futures scaled to 10% target volatility.
Figure 9: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to 10% target volatility.
Figure 10: Pairwise correlation between daily out-of-sample returns of portfolio strategies from Dec 1991 thru Nov 2016 in S&P500 Stocks scaled to full investment.
Figure 11 below reports the distribution of sharpe ratios of various portfolios over the entire study period when calibrated with 50 bootstrapped calibration periods. Calibration periods are generated by randomly drawing 36 month long blocks of returns out of the original 36-month calibration period.
Observations:
Figure 11: Distribution of out-of-sample sharpe ratios over entire study period when each portfolio strategy is calibrated with 50 bootstrapped calibration windows. Calibration periods are generated by randomly drawing 36 month long blocks of returns out of the original 36-month calibration period.
Exposure to low volatility assets is an important performance driver during the study period, during which low volatility assets earn a premium over high volatility assets due to falling interest rates and possibly other factors. At the same time, portfolios with a large exposure to low volatility assets are more exposed to the risk of rising interest rates in the future.
Figures 12 and 13 report the weighted average of single asset volatility held in each portfolio, where volatility \(\sigma_i\) of asset \(i\) is the standard deviation of the asset’s returns during the portfolio calibration period. \[ \text{avg. asset volatlitiy} = \frac{\mathbf{w}'\mathbf{\sigma_r}}{\mathbf{w}'\mathbf{1}} \]
Observations:
Figure 12: Average single asset volatility weighted by proportion held in portfolio. The plot is interactive, lines can be removed/restored by selecting the legend.
Figure 13: Average single asset volatility weighted by proportion held in portfolio. The plot is interactive, lines can be removed/restored by selecting the legend.
Figures 9-31 illustrate the portfolio’s allocations to underlying assets. For this illustration, portfolios are calibrated for a subset of 10 randomly selected instruments to keep the number of assets visually digestible.
The procedure to generate random subset avoids introducing a survivorship bias:
Observations:
Figures 14-21 report the fraction of funds allocated to underlying assets by the different portfolio strategies, figure 22 reports the rolling volatility of portfolio constituents during the calibration period.
Figure 14: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.
Figure 15: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.
Figure 16: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.
Figure 17: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.
Figure 18: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.
Figure 19: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.
Figure 20: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.
Figure 21: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.
Figure 22: Volatility of portfolio constituents during the rolling calibration period.
Figures 23-30 report the fraction of funds allocated to underlying assets by the different portfolio strategies, figure 31 reports the rolling volatility of portfolio constituents during the calibration period.
Figure 23: Fraction of funds allocated to underlying assets by the hierarchical risk parity with variance (HRP-VAR) portfolio.
Figure 24: Fraction of funds allocated to underlying assets by the hierarchical risk parity with volatility (HRP-VOL) portfolio.
Figure 25: Fraction of funds allocated to underlying assets by the equal weight (EW) portfolio.
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Figure 26: Fraction of funds allocated to underlying assets by the maximum sharpe ratio (MSR) portfolio.
Figure 27: Fraction of funds allocated to underlying assets by the inverse volatility (IV) portfolio.
Figure 28: Fraction of funds allocated to underlying assets by the global minimum variance (GMV) portfolio.
Figure 29: Fraction of funds allocated to underlying assets by the maximum diversified (MD) portfolio.
Figure 30: Fraction of funds allocated to underlying assets by the equal risk congtribution (ERC) portfolio.
Figure 31: Volatility of portfolio constituents during the rolling calibration period.
Choueifaty, Yves, and Yves Coignard. 2008. “Toward Maximum Diversification.” The Journal of Portfolio Management 35 (1): 40–51.
López de Prado, Marcos. 2016. “Building Diversified Portfolios That Outperform Out of Sample.” The Journal of Portfolio Management 42 (4): 59–69.
Maillard, Sébastien, Thierry Roncalli, and Jérôme Teïletche. 2010. “The Properties of Equally Weighted Risk Contribution Portfolios.” The Journal of Portfolio Management 36 (4): 60–70.